Topic 2.13 · HL only

Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $: notes and practice questions

Q: What does Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ cover in IB Maths AA?

Rational functions are ratios of polynomials, f(x) = P(x)/Q(x). Domain excludes roots of Q(x). Intercepts are found by setting Q(x)=0 (x-intercepts) and f(0) (y-intercept). For f(x) = (ax+b)/(cx^2+dx+e), the horizontal asymptote is y=0. Vertical asymptotes depend on the roots of the quadratic denominator.

Q: How do I revise Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ for IB Maths AA?

Start from the core idea: rational functions are ratios of polynomials, f(x) = P(x)/Q(x). In the exam: the two named forms are the whole permitted range, so a question with a quadratic over a quadratic is out of syllabus. The oblique asymptote is the distinguishing mark and it needs polynomial division shown. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $?

FourtyFive has 3 Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

3 min read Jump to 3 practice questions

Summary

Rational functions are ratios of polynomials, $f(x) = P(x)/Q(x)$ .
Domain excludes roots of $Q(x)$ . Intercepts are found by setting $Q(x)=0$ (x-intercepts) and $f(0)$ (y-intercept).
For $f(x) = \frac{ax+b}{cx^2+dx+e}$ , the horizontal asymptote is $y=0$ . Vertical asymptotes depend on the roots of the quadratic denominator.
For $f(x) = \frac{ax^2+bx+c}{dx+e}$ , a vertical asymptote exists at the root of the linear denominator, and an oblique asymptote is found via polynomial long division.
Integration often involves substitution, partial fractions, or polynomial division followed by term-by-term integration.
Calculus (first and second derivatives) is used for curve sketching, identifying stationary points, intervals of increase/decrease, concavity, and inflection points.

How it is examined

The two named forms are the whole permitted range, so a question with a quadratic over a quadratic is out of syllabus. The oblique asymptote is the distinguishing mark and it needs polynomial division shown. 5 to 8 marks.

Key ideas

Rational functions of the form $f(x) = \dfrac{ax+b}{cx^2+dx+e}$ and $f(x) = \dfrac{ax^2+bx+c}{dx+e}$ .

Linking questions

The second form is the one that produces an oblique asymptote, which is the only place oblique asymptotes appear in AA.

Practice questions

3 questions · 1 medium · 2 hard

Showing 3 of 3

Question 1

MediumPaper 1 · no calculator8 marks

(a)

Consider the function $g(x) = \frac{ax^2+x+k}{x-4}$ .

The graph of $y=g(x)$ passes through the point $(1, -2)$ and has an oblique asymptote with equation $y = -3x-11$ .

(a) Write down the equation of the vertical asymptote.

[1]

(b)(i)

(b) Find the value of:

(i) $a$

[2]

(b)(ii)

(ii) $k$

[2]

(c)

[3]

Question 2

HardPaper 2 · calculator20 marks

(a)

A civil engineer is analyzing the structural integrity of a new bridge design. The deflection of a certain point on the bridge, $D(x)$ , in millimeters, is modeled by the function $D(x) = \frac{2x+1}{x^2-4}$ , where $x$ represents the horizontal distance in meters from a central support. The model is valid for $x \in \mathbb{R}$ , $x\neq p$ , $x\neq q$ .

Find the value of $p$ and the value of $q$ .

[2]

(b)

Find an expression for $D'(x)$ .

[3]

(c)

The graph of $y = D(x)$ has exactly one point of inflexion.

Find the x-coordinate of the point of inflexion.

[2]

(d)

Sketch the graph of $y = D(x)$ for $-4 \leq x \leq 4$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima (if they exist), and giving the equations of any asymptotes.

[5]

(e)

Consider a related model for stress distribution, $S(x) = \frac{x^2-4}{2x+1}$ for $x \in \mathbb{R}$ , $x \neq -\frac{1}{2}$ .

Find the equations of all the asymptotes on the graph of $y = S(x)$ .

[4]

(f)

The engineer needs to identify the regions where the bridge deflection $D(x)$ is less than $1$ mm. Solve $D(x) < 1$ for $x \in \mathbb{R}$ .

[4]

Question 3

HardPaper 1 · no calculator8 marks

(a)

A biologist is modelling the population, P, of a species of wasp in a controlled environment. The population, in thousands, after t weeks is modelled by the function

$P(t) = \frac{10t^2 + at + b}{t+5}, \text{ where } t \ge 0$

The initial population is 8000 wasps.

(a) Use this information to find the value of b.

[2]

(b)

(b) For large values of t, the population growth can be approximated by the oblique asymptote with equation $P=10t-20$ . Find the value of a.

[3]

(c)

[3]

Every Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

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Using your own wrong value after failing a "show that".

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What does Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ cover in IB Maths AA?

Rational functions are ratios of polynomials, f(x) = P(x)/Q(x). Domain excludes roots of Q(x). Intercepts are found by setting Q(x)=0 (x-intercepts) and f(0) (y-intercept). For f(x) = (ax+b)/(cx^2+dx+e), the horizontal asymptote is y=0. Vertical asymptotes depend on the roots of the quadratic denominator.

Is Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ SL or HL?

Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ is HL only. SL students are not examined on it.

How do I revise Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ for IB Maths AA?

Start from the core idea: rational functions are ratios of polynomials, f(x) = P(x)/Q(x). In the exam: the two named forms are the whole permitted range, so a question with a quadratic over a quadratic is out of syllabus. The oblique asymptote is the distinguishing mark and it needs polynomial division shown. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $?

FourtyFive has 3 Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Harder rational functions of the form $ f(x) = \frac{ax+b}{cx^2+dx+e} $ and $ f(x) = \frac{ax^2+bx+c}{dx+e} $ answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Harder rational functions of the form \( f(x) = \frac{ax+b}{cx^2+dx+e} \) and \( f(x) = \frac{ax^2+bx+c}{dx+e} \): notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Every Harder rational functions of the form \( f(x) = \frac{ax+b}{cx^2+dx+e} \) and \( f(x) = \frac{ax^2+bx+c}{dx+e} \) question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

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How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Every Harder rational functions of the form \( f(x) = \frac{ax+b}{cx^2+dx+e} \) and \( f(x) = \frac{ax^2+bx+c}{dx+e} \) question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.